Answer
The solution is $(-2,0)\cup(2,\infty)$
The graph is
Work Step by Step
$x^{3}-4x\gt0$
Factor the left side:
$x(x^{2}-4)\gt0$
$x(x-2)(x+2)\gt0$
Find the intervals. The factors are $x$, $x-2$ and $x+2$. Set them equal to $0$ and solve for $x$:
$x=0$
$x-2=0$
$x=2$
$x+2=0$
$x=-2$
The factors are zero when $x=0,2,-2$. These three numbers divide the real line into the following intervals:
$(-\infty,-2)$ $,$ $(-2,0)$ $,$ $(0,2)$ $,$ $(2,\infty)$
Elaborate a diagram, using test points to determine the sign of each factor in each interval: (refer to the attached image below)
It can be seen from the diagram that the inequality is satisfied on the intervals $(-2,0)$ and $(2,\infty)$. Also, the inequality involves $\gt$ so the endpoints do not satisfy the inequality.
The solution is $(-2,0)\cup(2,\infty)$
